Prior dependence of cosmological constraints on dark matter-radiation interactions

We explore the issue of prior dependence in the context of one-sided constraints on the dark matter-photon and dark matter-neutrino elastic scattering cross-sections derived from cosmic microwave background (CMB) anisotropies measurements. Testing in particular the linear flat, Jeffreys, and logarithmic flat priors, we find that the former two yield upper limits on the cross-sections that are mutually consistent to within 20%. In contrast, bounds derived under the assumption of the logarithmic flat prior are strongly sensitive to the choice of the lower prior boundary. Indeed, surveying the recent literature, we find that this pathology of the logarithmic prior has resulted in published constraints that are up to an order of magnitude artificially tighter than they should objectively be. Our revised `objective' constraints from the 2015 data of the Planck CMB mission on the present-day scattering cross-sections are $\sigma_{\rm DM-\gamma} < 1.72 \times 10^{-6} \, \sigma_{\rm T} \, (m_{\rm DM}/{\rm GeV})$ and $\sigma_{\rm DM-\gamma} < 2.74 \times 10^{-15} \, \sigma_{\rm T} \, (m_{\rm DM}/{\rm GeV})$ for dark matter-photon interactions scaling as $a^0$ and $a^{-2}$ respectively, where $a$ is the scale factor, and $\sigma_{\rm T}$ the total Thomson scattering cross-section. Their dark matter-neutrino counterparts read $\sigma_{\rm DM-\nu} < 2.14 \times 10^{-6} \, \sigma_{\rm T} \, (m_{\rm DM}/{\rm GeV})$ and $\sigma_{\rm DM-\nu}< 2.46 \times 10^{-15} \, \sigma_{\rm T} \, (m_{\rm DM}/{\rm GeV})$. All have been computed assuming the Jeffreys prior.